We repeat similar arguments to [1]}.
Note that for any \(\lambda >0\)
\(\inf \limits _{(A,u)\in \mathcal {C},\,|A|=\mathtt {v}}\, \mathcal {F}(A,u)\,\, \ge \inf \limits _{(A,u)\in \widetilde{\mathcal {C}},\,|A|=\mathtt {v}}\, \widetilde{\mathcal {F}}(A,u) \ge \inf \limits _{(A,u)\in \widetilde{\mathcal {C}}}\, \widetilde{\mathcal {F}}^{\lambda }(A,u).\) 
Further we fix any \(\lambda \ge \lambda _0.\) 
Recall that from [1]} for such \(\lambda \)
\(\inf \limits _{(A,u)\in \mathcal {C},\,|A|=\mathtt {v}}\, \mathcal {F}(A,u) =\lim \limits _{m\rightarrow +\infty }\,\min \limits _{(A,u)\in \mathcal {C}_m,\,|A|=\mathtt {v}}\, \mathcal {F}(A,u)=\lim \limits _{m\rightarrow +\infty }\,\min \limits _{(A,u)\in \mathcal {C}_m}\, \mathcal {F}^{\lambda }(A,u),\) 
where \(\mathcal {F}^\lambda \)  is given by (REF ). Thus, in view of (REF ) to prove (REF ) it is enough to establish that \(\epsilon >0\)  there exists \(n_\epsilon \in \mathbb {N}\)  and \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon }\)  such that
\(\inf \limits _{(A,u)\in \widetilde{\mathcal {C}}}\, \widetilde{\mathcal {F}}^{\lambda }(A,u) + \epsilon > \mathcal {F}^{\lambda }(A_\epsilon ,u_\epsilon ).\) 
To prove the existence of \(n_\epsilon \)  and \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon }\) , we repeat essentially the same arguments of the proof of [1]}. For the convenience of the reader we give the detailed proof. Given \(\epsilon >0\)  let \((B_1,v_1)\in \widetilde{\mathcal {C}}\)  be such that
\(\inf \limits _{(A,u)\in \widetilde{\mathcal {C}}}\, \mathcal {F}^{\lambda }(A,u)> \mathcal {F}^{\lambda }(B_1,v_1) -\epsilon .\) 
Since \(|B_1| = |\mathrm {Int}(B_1)|\)  and \(\mathcal {F}^{\lambda }(B_1,v_1)\ge \mathcal {F}^{\lambda }(\mathrm {Int}(B_1),v_1),\)  we may assume that \(B_1=\mathrm {Int}(B_1),\)  i.e., \(B_1\)  is open.
Step 1: First we remove the jump set \(J_{v_1}\)  of \(v_1\)  on \(\Sigma \)  making a hole in \(\Omega .\)  Recall that by our choice, \(\nu _\Sigma \)  is always directed towards \(\Omega .\)  Since \(\Sigma \)  is Lipschitz, by the regularity of \(\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Sigma ,\)  there exists an open set \(\Sigma ^{\prime }\subset \Sigma \)  such that \(\mathcal {H}^1(J_{v_1}\setminus \Sigma ^{\prime })=0\)  and \(\mathcal {H}^1(\Sigma ^{\prime }\setminus J_{v_1})<\frac{\epsilon }{c_2}.\)
Let \(r\in (0,\frac{\epsilon }{\lambda \mathcal {H}^1(\Sigma )})\)  be such that
\(|\varphi (x,\nu ) - \varphi (y,\nu )|<\frac{\epsilon }{\mathcal {H}^1(\Sigma )}\) 
whenever \(|x-y|<4r.\)  Since \(\Sigma \)  is Lipschitz, by Vitaly Covering Lemma we can find an at most countable family \(\lbrace Q_{r_j,\nu _{\Sigma }(x_j)}(x_j)\rbrace _{j\ge 1}\)  of disjoint open squares such that \(x_j\in \Sigma ,\)  \(r_j\in (0,r),\)  \(\Sigma \cap Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  is a graph in \(\nu _\Sigma (x_j)\) -direction,
\(\Sigma \)  crosses two opposite sides of each \(Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  parallel to \(\nu _\Sigma (x_j)\)  and
\(\mathcal {H}^1\Big (\Sigma ^{\prime }\setminus \bigcup _j \overline{Q_{r_j,\nu _{\Sigma }(x_j)}(x_j)}\Big )=0.\) 
Note that \(\sum \limits _j r_j< \mathcal {H}^1(\Sigma ).\) 
For each \(j\)  define
\(\Sigma _j:=(\Sigma \cap \overline{Q_{r,\nu _{\Sigma }(x_j)}(x_j)}) + \rho _j\nu _\Sigma (x_j),\) 
where \(\rho _j\in (0,r_j)\)  is such that \(\Sigma _j\)  still connects two vertical sides of \(Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  and \(\sum _j \rho _j <\frac{\epsilon }{2c_2}.\)  Let \(U_j\)  be the
open set whose boundaries are \(\Sigma _j,\)  \(\Sigma \cap \overline{Q_{r,\nu _{\Sigma }(x_j)}(x_j)}\)  and two vertical sides of \(Q_{r,\nu _{\Sigma }(x_j)}(x_j).\)  Note that \(\lbrace U_j\rbrace _j\)  is a countable family of pairwise disjoint open sets.
Let \(B_2:=B_1\setminus \overline{\cup _j U_j}\)  and \(v_2:=v_1\Big |_{B_2\cup S}.\)  Then using the localized version of \(\mathcal {S}\)  we get
\(\mathcal {S}(B_2,v_2) \le & \mathcal {S}(B_1,v_1-u_0; \Omega \setminus \overline{\cup _j U_j}) + \sum \limits _{j} \Big (\int _{\Sigma _j} \varphi (x,\nu _\Sigma (x))d\mathcal {H}^1 + 2c_2 \rho _j\Big ).\) 

By (REF ) and the definition of \(\Sigma _j\)
\(\int _{\Sigma _j} \varphi (x,\nu _\Sigma (x))d\mathcal {H}^1 \le \int _{\Sigma \cap Q_{r,\nu _{\Sigma }(x_j)}(x_j)} \varphi (y,\nu _\Sigma (y))d\mathcal {H}^1 + \frac{\epsilon \mathcal {H}^1(\Sigma \cap Q_{r,\nu _{\Sigma }(x_j)}(x_j))}{\mathcal {H}^1(\Sigma )}.\) 
Thus summing this inequality in \(j\)  and using pairwise disjointness of \(Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  and (REF ) we get
\(\sum \limits _j \int _{\Sigma _j} \varphi (x,\nu _\Sigma (x))d\mathcal {H}^1 \le \int _{\Sigma ^{\prime }} \varphi (y,\nu _\Sigma (y))d\mathcal {H}^1 + \frac{\epsilon \mathcal {H}^1(\Sigma ^{\prime })}{\mathcal {H}^1(\Sigma )}.\) 
Using the definition of \(\Sigma ^{\prime }\)  we obtain
\(\sum \limits _j \int _{\Sigma _j} \varphi (x,\nu _\Sigma (x))d\mathcal {H}^1 \le \int _{J_{v_1}} \varphi (y,\nu _\Sigma (y))d\mathcal {H}^1+ 2\epsilon .\) 
Inserting this in (REF ) and using the inequality \(\sum _j \rho _j <\frac{\epsilon }{2c_2}\)  we get
\(\mathcal {S}(B_2,v_2) \le \mathcal {S}(B_1,v_1-u_0; \Omega \setminus \overline{\cup _j U_j}) + \int _{J_{v_1}} \varphi (y,\nu _\Sigma (y))d\mathcal {H}^1+ 3\epsilon \le \mathcal {S}(B_1,v_1) + 3\epsilon .\) 
Then by the nonnegativity of the elastic energy, for \((B_2,v_2)\)  we get
\(\widetilde{\mathcal {F}}(B_2,v_2) \le \widetilde{\mathcal {F}}(B_1,v_1) + 3\epsilon .\) 
Notice that by our construction \(\Sigma \cap J_{v_2}\)  is \(\mathcal {H}^1\) -negligible, hence by Lemma REF  \(v_2\in H^1_\mathrm {loc}({\rm Int}{(B_2\cup S\cup \Sigma )}).\)
Finally we estimate the volume contribution of \(B_2.\)  Since \(U_j\subset Q_{r,\nu _{\Sigma }(x_j)}(x_j)\)  and
\(r_j\le r<\frac{\epsilon }{\lambda \mathcal {H}^1(\Sigma )},\)  using \(\sum _j r_j <\mathcal {H}^1(\Sigma )\)  we get
\(|B_1\setminus B_2| \le \sum \limits _j |U_j| \le \sum \limits _j r_j^2 \le r\sum \limits _j r_j < \frac{\epsilon }{\lambda }.\) 
Therefore,
\(\widetilde{\mathcal {F}}^{\lambda }(B_1,v_1) \ge \widetilde{\mathcal {F}}^{\lambda }(B_2,v_2) - 4\epsilon .\) 
Step 2: Let \(\lbrace E_i\rbrace _{i\ge 1}\)  be all open connected components of \(B_2\)  (recall that \(B_2\)  is open). We remove all sufficiently small connected components of \(B_1.\)  Using the localized versions of \(\mathcal {S}\)  and \(\mathcal {W}\)  we have
\(\mathcal {W}(B_2,v_2-u_0;\Omega ) = \sum \limits _{i\ge 1} \mathcal {W}(E_i,v_2-u_0;\Omega ).\) 
Since \(\partial E_i\cap \partial E_j\subset B_2^{(1)}\cap \partial B_2\)  and \(\varphi (x,\cdot )\)  is even,
\(\mathcal {S}(B_2,v_2;\Omega ) = \sum \limits _{i\ge 1} \mathcal {S}(E_i,v_2;\Omega ).\) 
Hence, there exists \(N_1\in \mathbb {N}\)  such that the set
\(B_3:=\cup _{i=1}^{N_1} E_i\) 
satisfies
\({\begin{array}{c}\mathcal {S}(B_2,v_2;\Omega ) +\mathcal {W}(B_2,v_2-u_0;\Omega )+ \epsilon >\mathcal {S}(B_3,v_2;\Omega )+\mathcal {W}(B_3,v_2-u_0;\Omega ),\\0 \le |B_2| - |B_3| < \frac{\epsilon }{\lambda }.\end{array}}\) 
Thus,
\(\mathcal {F}^{\lambda }(B_2,v_2) > \mathcal {F}^{\lambda }(B_3,v_3) - 2\epsilon ,\) 
where \(v_3:=v_2\Big |_{B_3}.\)
Step 3: Let \(\lbrace F_j\rbrace _{j\ge 1}\)  be all connected components of \(\Omega \setminus \overline{B_3}\)  such that \(\partial F_j\subset \partial B_3\)  (hence, \(F_i\)  are holes in \(B_3\) ). We fill in all sufficiently small holes. Since \(\mathcal {S}(B_3,v)<+\infty ,\)  there exists \(N_2\ge 1\)  such that
\(\sum \limits _{i>N_2} \mathcal {S}(F_i,v_3;\Omega ) + \sum \limits _{i>N_2} \mathcal {W}(F_i,v_3-u_0;\Omega ) <\epsilon ,\quad \sum \limits _{i>N_2} |F_i|<\frac{\epsilon }{\lambda }.\) 
Then the set \(B_4:= B_3\cup (\cup _{i>N_2} F_i)\)  and the function \(v_4:=v_3\chi _{B_2\cup S} + u_0\chi _{\cup _{i>N_2} F_i}\)  satisfies
\(\mathcal {F}^{\lambda }(B_3,v_3)>\mathcal {F}^{\lambda }(B_4,v_4) - 2\epsilon .\) 
By construction, \(\overline{\partial ^* B_4}\)  has at most \(N_1 + N_2\)  connected components.
Step 5: Finally we construct \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon }\)  satisfying (REF ) for some \(n_\epsilon \in \mathbb {N}.\)  Let \(B_5:=\mathrm {Int}(\overline{B_4}).\)  Since \(B_4\)  can have finitely many “substantial” holes \(B_5\cap \partial B_4=\emptyset .\)  In particular, if we extend \(v_4\)  arbitrarily to the set \(B_4^{(1)}\cap \partial B_4\)  and denote the extension by \(v_5,\)  then \(v_5\in GSBD^2({\rm Int}{(B_5\cup S\cup \Sigma )})\)  and \(J_{v_5}=S_{v_4}^{B_4}\)  up to a \(\mathcal {H}^1\) -negligible set, where \(S_u^A\)  is defined in (REF ).
Since \(v_5=v_4\)  a.e. in \(B_5,\)  by (REF )-(REF )
\(\int _{B_5\cup S} x)e(v_5):e(v_5) = \mathcal {W}(B_4,v_4) \le \widetilde{\mathcal {F}}(B_4,v_4) + c_2\mathcal {H}^1(\Sigma ) \le C+ 9\epsilon ,\) 
where \(C:=\max \lbrace 1,\inf _{\widetilde{\mathcal {C}}}\widetilde{\mathcal {F}}\rbrace \)  is independent of \(\epsilon .\)
By [4]} there exists \(u_\epsilon \in SBV^2({\rm Int}{(B_5\cup S\cup \Sigma )})\cap L^\infty ({\rm Int}{(B_5\cup S\cup \Sigma )})\)  such that \(J_{u_\epsilon }\)  is contained in a union \(\Gamma \)  of finitely many closed connected pieces of \(C^1\) -curves in \({\rm Int}{(B_5\cup S\cup \Sigma )},\)  \(u_\epsilon \in W^{1,\infty }({\rm Int}{(B_5\cup S\cup \Sigma )})\)  and
\(\int _{B_5\cup S} |e(u_\epsilon ) - e(v_5)|^2 dx \le \frac{\epsilon }{4(C+11\epsilon )(\Vert _\infty +1)}\) 
and
\(\mathcal {H}^1(J_{u_\epsilon }\Delta J_{v_5})<\frac{\epsilon }{2c_2}.\) 
Since \(J_{v_5}\subset B_5,\)  we can assume that the squares \(\lbrace Q_j\rbrace _{j\ge 1}\)  of Vitali cover in [4]} satisfies \(Q_j\subset \subset B_5.\)  Therefore, we may assume that \(\Gamma \subset \overline{B_5}.\)  Since \(\mathcal {H}^1\mathop {\hbox{\vrule height 9pt width 0.5pt depth 0pt\vrule height 0.5pt width 5pt depth 0pt}}\nolimits \Gamma \)  is regular, we may extract finitely many intervals of \(\Gamma \)  whose union \(\Gamma ^{\prime }\)  still covers \(J_{u_\epsilon }\)  and satisfies \(\mathcal {H}^1(\Gamma ^{\prime }\setminus J_{u_\epsilon })<\frac{\epsilon }{2c_2}.\) 
Now we define
\(A_\epsilon :=B_5\setminus \overline{\Gamma ^{\prime }}.\) 
Recall that both \(\Sigma \cap J_{v_5}\)  and \(\Sigma \cap J_{e-\epsilon }\)  are \(\mathcal {H}^1\) -negligible.
By the definition of \(B_5\)  and \(\Gamma ^{\prime },\)  there exists \(n_\epsilon \in \mathbb {N}\)  such that \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon }.\)  By the definition of \(\widetilde{\mathcal {S}},\)  \(B_5\)  and \(v_5\)  as well as by (REF ) we have
\(\widetilde{\mathcal {S}}(B_4,v_4) = & \int _{\Omega \cap \partial ^* B_5} \varphi (x,\nu _{B_5})d\mathcal {H}^1 + 2\int _{B_5\cap J_{v_5}} \varphi (x,\nu _{J_{v_5}})d\mathcal {H}^1 + \int _{\Sigma \cap \partial ^*B_5} \beta d\mathcal {H}^1\nonumber \\\ge & \int _{\Omega \cap \partial ^* B_5} \varphi (x,\nu _{B_5})d\mathcal {H}^1 + 2\int _{B_5\cap J_{u_\epsilon }} \varphi (x,\nu _{J_{u_\epsilon }})d\mathcal {H}^1 + \int _{\Sigma \cap \partial ^*B_5} \beta d\mathcal {H}^1 - \epsilon .\) 

Thus, by the definition of \(A_\epsilon \)  and \(\Gamma ^{\prime }\)
\( \widetilde{\mathcal {S}}(B_4,v_4) \ge & \int _{\Omega \cap \partial ^* A_\epsilon } \varphi (x,\nu _{A_\epsilon })d\mathcal {H}^1 + 2\int _{A_\epsilon ^{(1)}\cap \Gamma ^{\prime }} \varphi (x,\nu _{\Gamma ^{\prime }})d\mathcal {H}^1 + \int _{\Sigma \cap \partial ^*A_\epsilon } \beta d\mathcal {H}^1 - 2\epsilon \nonumber \\= & \mathcal {S}(A_\epsilon ,u_\epsilon ) - 2\epsilon .\) 

Moreover, using the relations \(|A_\epsilon \Delta B_4|=0\)  and \(v_4=v_5\)  a.e. in \(B_5\)  and Cauchy-Schwartz inequality for nonnegative symmetric forms we obtain
\( \mathcal {W}(A_\epsilon ,u_\epsilon ) \le & \mathcal {W}(B_4,v_4) + 2\int _{B_5\cup S} x)e(u_\epsilon ):(e(u_\epsilon ) - e(v_5))\nonumber \\\le & \mathcal {W}(B_4,v_4) + 2\Big (\int _{B_5\cup S} x)e(u_\epsilon ):e(u_\epsilon )dx\Big )^{1/2}\times \nonumber \\&\times \Big (\int _{B_5\cup S} x) (e(u_\epsilon ) - e(v_5)):(e(u_\epsilon ) - e(v_5))dx\Big )^{1/2}.\) 

Similarly,
\(& \int _{B_4\cup S} x)e(u_\epsilon ):e(u_\epsilon )dx\\\le & \mathcal {W}(B_4,v_4) + 2\Big (\mathcal {W}(B_4,v_4)\Big )^{1/2}\Big (\int _{B_5\cup S} x) (e(u_\epsilon ) - e(v_5)):(e(u_\epsilon ) - e(v_5))dx\Big )^{1/2}\\\le & (C+9\epsilon ) + 2\sqrt{(C+9\epsilon )\Vert _\infty }\,\,\Vert e(u_\epsilon ) - e(v_5)\Vert _{L^2} \le C+10\epsilon ,\) 

where in the last inequality we used (REF ).
Therefore, by (REF ) and again by (REF )
\(\mathcal {W}(A_\epsilon ,u_\epsilon ) \le & \mathcal {W}(B_4,v_4) + 2\sqrt{(C+10\epsilon )\Vert _\infty }\,\,\Vert e(u_\epsilon ) - e(v_5)\Vert _{L^2} \le \mathcal {W}(B_4,v_4) +\epsilon .\) 

Now combining (REF ) and (REF ) as well as using \(|B_5|=|A_\epsilon |\)  we get
\(\widetilde{\mathcal {F}}^{\lambda }(B_4,v_4) \ge \mathcal {F}^{\lambda }(A_\epsilon ,u_\epsilon ) - 3\epsilon .\) 
Since \((A_\epsilon ,u_\epsilon )\in \mathcal {C}_{n_\epsilon },\)  by (REF ), (REF ), (REF ), (REF ) and (REF ) we get
\(\inf \limits _{(A,u)\in \widetilde{\mathcal {C}}} \,\widetilde{\mathcal {F}}(A,u) + 12 \epsilon \ge \mathcal {F}(A_\epsilon ,u_\epsilon ),\) 
and (REF ) follows.
